1. Field of the Invention
The present invention relates to a mask fabrication method, exposure method, device fabrication method, and recording medium.
2. Description of the Related Art
An exposure apparatus has conventionally been used when fabricating a fine semiconductor device such as a semiconductor memory or logic circuit by using photolithography. In the exposure apparatus, a projection optical system projects patterns (circuit patterns) formed on a mask (reticle) onto a substrate such as a wafer, thereby transferring the patterns. Recently, the exposure apparatus is used to form patterns having dimensions smaller than the exposure wavelength (the wavelength of exposure light). In the resolution of such fine patterns, the image performance changes in accordance with illumination conditions (an effective source) by which a mask is illuminated. Therefore, it is important to set an optimum effective source. Ordinary, the optimization of the effective source generally uses the calculation of an optical image (aerial image). For example, a light source is two-dimensionally divided into a plurality of elements, and an element is regarded as point light source. A point source illuminates a mask, and the diffracted light from mask pattern is resolved on an image plane through the optical system. Then an optical image from the source is calculated, and a point source which contributes to the resolution of a pattern image is selected. This makes it possible to optimize the effective source (see Japanese Patent Laid-Open No. 6-120119). However, this method takes an enormous time because the optical image (aerial image) is to be calculated.
Also, since a pattern having a dimension smaller than the exposure wavelength, the pattern shape is not transferred on a wafer with accuracy. More specifically, the corner of the pattern is rounded, or the length of the pattern decreases. This largely decreases the shape accuracy of a pattern formed on a wafer.
To reduce the decrease in shape accuracy of a pattern formed on a wafer, therefore, a mask pattern is designed by performing the process of correcting the pattern shape. Also, a mask pattern is sometimes designed by performing the process of inserting an assist pattern having a dimension not to be resolved in a main pattern to be resolved on a wafer. These processes are called optical proximity correction (OPC).
The size of an image of a mask pattern is normalized by the numerical aperture (NA) of a projection optical system or a wavelength λ of exposure light, and represented by a k1(=HP×NA/λ) factor in which HP is the half pitch of the mask pattern. Recently, the k1 factor is approaching the resolution limit (k1=about 0.25) of lithography. However, the optical proximity effect increases as decreases the k1 factor, and this makes OPC very important.
OPC is generally automatically performed by a computer. When correcting a pattern shape, for example, the pattern shape is corrected for each element of the mask pattern by a rule base or a model base using optical simulation, by taking account of the shape of the element and the influence of surrounding elements.
In the model base using optical simulation, a mask pattern is deformed until a target pattern is obtained, and various methods are usable as the method of deformation. An example is an iterative improvement method by which if an optical image is partially expanded, a mask pattern is narrowed by the some amount, and if the optical image is partially narrowed, the mask pattern is expanded by the some amount. The mask pattern is thus gradually deformed while the optical image is recalculated. A method of deforming a mask pattern by using a genetic algorithm has also been proposed. However, these methods use an enormous time because the optical image (aerial image) is to be calculated a number of times until a desired pattern is obtained.
On the other hand, for the method of inserting an assist pattern in a main pattern, a technique that derives how to insert the assist pattern by numerical calculations has been disclosed (see Japanese Patent Laid-Open No. 2004-221594). In this technique, an interference map is obtained by numerical calculations, and an interference position (region) and interference cancellation position (region) on a mask are derived. In the interference position on the interference map, an assist pattern that equalizes the phase of light having passed through a main pattern and the phase of light having passed through the assist pattern is inserted. Consequently, the light having passed through the main pattern and the light having passed through the assist pattern strongly interfere with each other. This makes it possible to accurately form a target pattern on a wafer. Note that the mask surface and wafer surface have an image formation relationship, so the interference map can also be regarded as obtaining the amplitude on the image plane. Furthermore, the target pattern is an element existing on a mask and to be transferred onto a wafer.
In addition, Japanese Patent Laid-Open No. 2008-040470 has disclosed a method of numerically obtaining information of an assist pattern.
The relationship between a mask pattern and wafer pattern in an exposure apparatus is a partial coherent image formation relationship. In the partial coherent image formation, an aerial image can be calculated by obtaining the coherence on the mask surface from information of an effective source, and performing Fourier integration on the coherence and the spectral distribution (diffracted light distribution) of a mask pattern. The “coherence” herein mentioned is the degree of interference corresponding to the distance on the mask surface. Also, the “effective source” is a light intensity distribution formed on the pupil plane of a projection optical system without any mask.
The coherence of the effective source can be considered by using a transmission cross coefficient (TCC). The TCC is defined by the pupil plane of a projection optical system, and is the overlapped portion of the effective source, the pupil function of the projection optical system, and the complex conjugate of the pupil function of the projection optical system.
In Japanese Patent Laid-Open No. 2004-221594, the TCC function is two-dimensionally expressed by fixing the position of the pupil, thereby obtaining an approximated aerial image (to be referred to as an approximate aerial image hereinafter). From the approximate aerial image, an assist pattern is placed near a peak position expect for a pattern to be resolved.
The interference map of Japanese Patent Laid-Open No. 2004-221594 forms an aerial image when squared, and hence can be regarded as a kind of an approximate aerial image.
Since OPC as described above depends on an effective source, OPC is generally performed after an effective source is set. When OPC is performed, however, a pattern deforms or an assist pattern is inserted. Accordingly, the effective source set before OPC may not be optimum any longer.
Especially when setting an assist pattern for an initially effective source by the method as described above, it is sometimes difficult to insert the assist pattern, or the image performance does not improve in some cases even when the assist pattern is inserted.
This is so because the resolution performance (e.g., the contrast or depth) with respect to the pitch between pattern elements is decided by a light source. In other words, the position of interference with a main pattern is uniquely decided by a light source. In the methods disclosed in Japanese Patent Laid-Open Nos. 2004-221594 and 2008-040470, if there is no position of interference with a main pattern, it is determined that no assist pattern is to be inserted, so no assist pattern is inserted. In this case, the resolution performance degrades if an assist pattern is forcibly inserted.
Accordingly, if a light source is unsuited to resolve the pitches of a main pattern and assist pattern, the assist pattern cannot be inserted, and this makes it difficult to improve the image formation performance.
Unfortunately, no optimum light source can be obtained if the pitches of a main pattern and assist pattern and a direction connecting the main pattern position and assist pattern position are unknown.
Since an optimum effective source and optimum mask pattern are closely related as described above, an optimum combination of the effective source and mask pattern is obtained to improve the image formation performance.